a revolving light is located at 5 km from a straight shoreline with a constant angular velocity. what velocity does the light revolve if the light moves along the shoreline at the rate of 15km/min when the beam of light makes a angle of 60 with the shoreline?

To find the angular velocity of the revolving light, we need to use the formula:

Angular Velocity = Linear Velocity / Radius

Given:

Linear Velocity (V) = 15 km/min

Radius (r) = 5 km

We need to convert the Linear Velocity to m/min for consistency:

Linear Velocity (V) = 15 km/min × 1000 m/km = 15000 m/min

Substitute the values into the formula:

Angular Velocity = 15000 m/min / 5 km

Note that 1 km = 1000 m, so we need to convert km to meters:

Angular Velocity = 15000 m/min / (5 km × 1000 m/km)

Angular Velocity = 15000 m/min / 5000 m

Angular Velocity = 3 rad/min

Therefore, the angular velocity of the revolving light is 3 radians per minute.

To find the velocity at which the light revolves, we can use trigonometry. Let's break down the problem step by step:

1. First, let's visualize the problem. We have a revolving light located 5 km from a straight shoreline. The light beam moves along the shoreline at a rate of 15 km/min when the beam makes an angle of 60° with the shoreline.

2. The velocity we need to find is the speed at which the light revolves, but since we know the beam is moving along the shoreline, this velocity is the component of the light's velocity perpendicular to the shoreline.

3. To find this component, we need to use trigonometry. In this case, we'll be using the sine function. Recall that the sine of an angle is equal to the opposite side divided by the hypotenuse in a right triangle. In this scenario, the opposite side is the velocity we want to find, and the hypotenuse is the 15 km/min velocity along the shoreline.

4. Let's assign variables to our known values. The velocity we want to find perpendicular to the shoreline will be represented as v (in km/min), and the velocity along the shoreline is given as 15 km/min.

5. Using the formula for sine, we have sin(60°) = v / 15. Rearranging the formula, we get v = 15 * sin(60°).

6. Now we need to evaluate sin(60°). By using a calculator or reference table, we find that sin(60°) = √3/2.

7. Plugging in the value, we get v = 15 * (√3/2).

8. Simplifying the expression, v = 7.5√3 km/min.

this is exactly analogous to your other problem with the revolving spotlight. See what you can do for it. Here, you are solving for dθ/dt instead of dx/dt.